## Second order algebraic knot concordance group

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##### Date

2011-06-28##### Author

Powell, Mark Andrew

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Let Knots be the abelian monoid of isotopy classes of knots S1 ⊂ S3 under connected sum, and let C be the topological knot concordance group of knots modulo slice knots. Cochran-Orr-Teichner [COT03] defined a filtration of C:
C ⊃ F(0) ⊃ F(0.5) ⊃ F(1) ⊃ F(1.5) ⊃ F(2) ⊃ . . .The quotient C/F(0.5) is isomorphic to Levine’s algebraic concordance group AC1 [Lev69]; F(0.5)
is the algebraically slice knots. The quotient C/F(1.5) contains all metabelian concordance
obstructions. The Cochran-Orr-Teichner (1.5)-level two stage obstructions map the concordance
class of a knot to a pointed set (COT (C/1.5),U).
We define an abelian monoid of chain complexes P, with a monoid homomorphism Knots →
P. We then define an algebraic concordance equivalence relation on P and therefore a group
AC2 := P/ ~, our second order algebraic knot concordance group. The results of this thesis can
be summarised in the following diagram: . That is, we define a group homomorphism C → AC2 which factors through C/F(1.5). We can
extract the two stage Cochran-Orr-Teichner obstruction theory from AC2: the dotted arrows
are morphisms of pointed sets. Our second order algebraic knot concordance group AC2 is a
single stage obstruction group.